Finite type morphisms with discrete fibers are quasi-finite.

Lemma 29.20.8. Let $f : X \to S$ be a morphism of schemes. Assume $f$ is locally of finite type. Then the following are equivalent

1. $f$ is locally quasi-finite,

2. for every $s \in S$ the fibre $X_ s$ is a discrete topological space, and

3. for every morphism $\mathop{\mathrm{Spec}}(k) \to S$ where $k$ is a field the base change $X_ k$ has an underlying discrete topological space.

Proof. It is immediate that (3) implies (2). Lemma 29.20.6 shows that (2) is equivalent to (1). Assume (2) and let $\mathop{\mathrm{Spec}}(k) \to S$ be as in (3). Denote $s \in S$ the image of $\mathop{\mathrm{Spec}}(k) \to S$. Then $X_ k$ is the base change of $X_ s$ via $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(\kappa (s))$. Hence every point of $X_ k$ is closed by Lemma 29.20.4. As $X_ k \to \mathop{\mathrm{Spec}}(k)$ is locally of finite type (by Lemma 29.15.4), we may apply Lemma 29.20.6 to conclude that every point of $X_ k$ is isolated, i.e., $X_ k$ has a discrete underlying topological space. $\square$

Comment #3032 by Brian Lawrence on

Suggested slogan: A locally finite-type morphism is locally quasi-finite if and only if its fibres are discrete.

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